If the infinite sum 12^1+ 12^2+ 12^3+ 12^4+...=1, what is the value of the infinite sum 12^1+ 22^2+ 32^3+ 42^4+....?

If the infinite sum 12^1+ 12^2+ 12^3+ 12^4+..… | GMAT Sprint

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705-805

Sequences

If the infinite sum $\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...=1$ , what is the value of the infinite sum $\frac{1}{2^1}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+....$ ?

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Infinite

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Official Explanation

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Given:

\frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + ... = 1

This can be written as:

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + ... = 1

............(I)

Note that the first term is

\frac{1}{2}

, so the sum of the rest of the terms must be

\frac{1}{2}

to add up to a total of 1.

So:

\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + ... = \frac{1}{2}

Similarly, now the first term is

\frac{1}{4}

, so the sum of the rest of the terms must be

\frac{1}{4}

to get a sum of

\frac{1}{2}

, and so on.

Required sum:

\frac{1}{2^1} + \frac{2}{2^2} + \frac{3}{2^3} + \frac{4}{2^4} + ....

…

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What this question tests

What it tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Common trap

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.

If the infinite sum 121+122+123+124+...=1\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...=1211​+221​+231​+241​+...=1, what is the value of the infinite sum 121+222+323+424+....\frac{1}{2^1}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+....211​+222​+233​+244​+....?